OFFSET

1,5

COMMENTS

Note that A002375 (which differs only at the n = 2 term) is the main entry for this sequence.

The graph of this sequence is called Goldbach's comet. - David W. Wilson, Mar 19 2012

The Goldbach conjecture states that a(n) > 0 for n >= 2. - Wolfdieter Lang, May 14 2016

With the second Maple program, the command G(2n) yields all the unordered pairs of prime numbers having sum 2n; caveat: a pair {a,a} is listed as {a}. Example: G(26) yields {{13}, {3,23}, {7,19}}. The command G(100000) yields 810 pairs very fast. - Emeric Deutsch, Jan 03 2017

Conjecture: Let p denote any prime in any decomposition of 2n. 4 and 6 are the only numbers n such that 2n + p is prime for every p. - Ivan N. Ianakiev, Apr 06 2017

Conjecture: For all m >= 0, there exists at least one possible value of n such that a(n) = m. - Ahmad J. Masad, Jan 06 2018

The previous conjecture is related to the sequence A053033. - Ahmad J. Masad, Dec 09 2019

Conjecture: For each k >= 0, there exists a minimum sufficiently large number r that depends on k such that for each n >= r, a(n) > k. - Ahmad J. Masad, Jan 08 2020

Conjecture: If the previous conjecture is true, then for each m >= 0, the number of terms that are equal to (m+1) is larger than the number of terms that are equal to m. - Ahmad J. Masad, Jan 08 2020

REFERENCES

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, MA, 1996, Chapter 12, pages 236-257.

H. Halberstam and H. E. Richert, 1974, "Sieve methods", Academic press, London, New York, San Francisco.

LINKS

H. J. Smith, Table of n, a(n) for n = 1..20000

M. Herkommer, Goldbach Conjecture Research.

Eric Weisstein's World of Mathematics, Goldbach Partition.

Wikipedia, Goldbach's conjecture.

G. Xiao, WIMS server, Goldbach

FORMULA

From Halberstam and Richert: a(n) < (8+0(1))*c(n)*n/log(n)^2 where c(n) = Product_{p>2} (1 - 1/(p-1)^2)*Product_{p|n, p>2} (p-1)/(p-2). It is conjectured that the factor 8 can be replaced by 2. - Benoit Cloitre, May 16 2002

a(n) = Sum_{i=2..n} floor(2/Omega(i*(2*n-i))). - Wesley Ivan Hurt, Jan 24 2013

a(n) = A224709(n) + (primepi(2n-2) - primepi(n-1)) + primepi(n) + 1 - n. - Anthony Browne, May 03 2016

MAPLE

A045917 := proc(n)

local a, i ;

a := 0 ;

for i from 1 to n do

if isprime(i) and isprime(2*n-i) then

a := a+1 ;

end if;

end do:

a ;

end proc: # R. J. Mathar, Jul 01 2013

# second Maple program:

G := proc (n) local g, j: g := {}: for j from 2 to (1/2)*n do if isprime(j) and isprime(n-j) then g := `union`(g, {{n-j, j}}) end if end do: g end proc: seq(nops(G(2*n)), n = 1 .. 98); # Emeric Deutsch, Jan 03 2017

MATHEMATICA

f[n_] := Length[Select[2n - Prime[Range[PrimePi[n]]], PrimeQ]]; Table[ f[n], {n, 100}] (* Paul Abbott, Jan 11 2005 *)

nn = 10^2; ps = Boole[PrimeQ[Range[1, 2*nn, 2]]]; Join[{0, 1}, Table[Sum[ps[[i]] ps[[n-i+1]], {i, Ceiling[n/2]}], {n, 3, nn}]] (* T. D. Noe, Apr 13 2011 *)

PROG

(PARI) a(n)=my(s); forprime(p=2, n, s+=isprime(2*n-p)); s \\ Charles R Greathouse IV, Mar 27 2012

(Haskell)

a045917 n = sum $ map (a010051 . (2 * n -)) $ takeWhile (<= n) a000040_list

-- Reinhard Zumkeller, Sep 02 2013

(Python)

from sympy import isprime

def A045917(n):

x = 0

for i in range(2, n+1):

if isprime(i) and isprime(2*n-i):

x += 1

return x # Chai Wah Wu, Feb 24 2015

(Magma) [#RestrictedPartitions(2*n, 2, Set(PrimesInInterval(1, 2*n))):n in [1..100]]; // Marius A. Burtea, Jan 23 2020

CROSSREFS

Cf. A002375 (the main entry for this sequence (which differs only at the n=2 term)).

KEYWORD

AUTHOR

STATUS

approved